The number of directions determined by less
than q points
Szabolcs L. Fancsali, Peter Sziklai and Marcella Takáts∗
February 28, 2012
Abstract
In this article we prove a theorem about the number of directions determined by less then q affine points, similar to the result of
Blokhuis et. al. [3] on the number of directions determined by q affine
points.
1
Introduction
In this article, p is a prime and q = ph , where h ≥ 1. GF(q) denotes the
finite field with q elements, and F can denote an arbitrary field (or maybe
a Euclidean ring). PG(d, q) denotes the projective geometry of dimension d
over the finite field GF(q). AG(d, q) denotes the affine geometry of dimension
d over GF(q) that corresponds to the co-ordinate space GF(q)d of rank d over
GF(q).
For the affine and projective planes AG(2, q) ⊂ PG(2, q), we imagine the
line ℓ∞ = PG(2, q) \ AG(2, q) at infinity as the set ℓ∞ ∼
= GF(q) ∪ {∞}. So the
non-vertical directions are field-elements (numbers) and the vertical direction
is ∞.
The original problem of direction sets was the following. Let f : GF(q) →
GF(q) be a function and let U = {(x, f (x)) | x ∈ GF(q)} ⊆ AG(2, q) be
the graph of the function f . The question is, how many directions can be
determined by the graph of f .
The authors were partially supported by the following grants: OTKA K 81310 and
OTKA CNK 77780, ERC, Bolyai and TÁMOP 4.2.1./B-09/KMR-2010-0003
∗
1
Definition (Direction set). If U denotes an arbitrary set of points in the
affine plane AG(2, q) then we say that the set
b − d (a, b), (c, d) ∈ U, (a, b) 6= (c, d)
D=
a−c a
as ∞ if a 6= 0, thus
0
D ⊆ GF(q) ∪ {∞}. If U is the graph of a function, then it simply means that
|U| = q and ∞ ∈
/ D.
is the set of directions determined by U. We define
In [1], Simeon Ball proved a stronger version of the structure theorem
of Aart Blokhuis, Simeon Ball, Andries Brouwer, Leo Storme and Tamás
Szőnyi, published in [3]. To recall their result we need some definitions.
Definition. Let U be a set of points of AG(2, q). If y ∈ ℓ∞ is an arbitrary
direction, then let s(y) denote the greatest power of p such that each line ℓ
of direction y meets U in zero modulo s(y) points. In other words,
s(y) = gcd |ℓ ∩ U| ℓ ∩ ℓ∞ = {y} ∪ ph .
Let s be the greatest power of p such that each line ℓ of direction in D meets
U in zero modulo s points. In other words,
s = gcd s(y) = min s(y).
y∈D
y∈D
Note that s(y) and thus also s might be equal to 1. Note that s(y) = 1 for
each non-determined direction y ∈
/ D.
Remark 1. Suppose that s ≥ p. Then for each line ℓ ⊂ PG(2, q):
either
or
(U ∪ D) ∩ ℓ = ∅;
|(U ∪ D) ∩ ℓ| ≡ 1 (mod s).
Moreover, |U| ≡ 0 (mod s).
(If s = 1 then 0 ≡ 1 (mod s), so in this case these remarks above would be
meaningless.)
Proof. Fix a direction y ∈ D. Each affine line with slope y meets U in zero
modulo s points, so |U| ≡ 0 (mod s).
An affine line L ⊂ AG(2, q) with slope y ∈ D meets U in 0 (mod s) points,
so the projective line ℓ = L ∪{y} meets U ∪ D in 1 (mod s) points.
An affine line L ⊂ AG(2, q) with slope y ∈
/ D meets U in at most one point,
so the projective line ℓ = L ∪{y} meets U ∪ D in either zero or one point.
2
Let P ∈ U and let L ⊂ AG(2, q) an affine line with slope y ∈ D, such that
P ∈ L. Then the projective line ℓ = L ∪{y} meets U in 0 (mod s) points,
and thus, ℓ meets D ∪ U \ {P } also in 0 (mod s) points. Thus, considering
all the lines through P (with slope in D), we get |U ∪ D| ≡ 1 (mod s). Since
U has 0 (mod s) points, |D| ≡ 1 (mod s). So we get that U ∪ D meets also
the ideal line in 1 (mod s) points.
Remark 2 (Blocking set of Rédei type). If |U| = q then each of the q affine
lines with slope y ∈
/ D meets U in exactly one point, so B = U ∪ D is a
blocking set meeting each projective line in 1 (mod s) points. Moreover, if
∞∈
/ D then U is the graph of a function, and in this case the blocking set B
above is called of Rédei type.
Theorem 3 (Blokhuis, Ball, Brouwer, Storme and Szőnyi; and Ball). [3]
and [1, Theorem 1.1] Let |U| = q and ∞ ∈
/ D. Using the notation s defined
above, one of the following holds:
either
s=1
and
or
GF(s) is a subfield of GF(q)
and
or
s=q
and
q+3
≤ |D| ≤ q;
2
q
q−1
+ 1 ≤ |D| ≤
;
s
s−1
|D| = 1.
Moreover, if s > 2 then U is a GF(s)-linear affine set (of rank logs q).
Definition (Affine linear set). A GF(s)-linear affine set is the GF(s)-linear
span of some vectors in AG(n, q) ∼
= GF(slogs q )n ∼
= GF(s)n logs q (or possibly a
translate of such a span). The rank of the affine linear set is the rank of this
span over GF(s).
What about the directions determined by an affine set U ⊆ AG(2, q) of
cardinality not q? Using the pigeon hole principle, one can easily prove that
if |U| > q then it determines all the q + 1 directions. So we can restrict our
research to affine sets of less than q points.
Examining the case q = p prime, Tamás Szőnyi [7] and later (independently) also Aart Blokhuis [2] have proved the following result.
Theorem 4 (Szőnyi; Blokhuis). [7, Theorem 5.2] Let q = p prime and
suppose that 1 < |U| ≤ p. Also suppose that ∞ ∈
/ D. Then
|U| + 3
≤ |D| ≤ p;
2
and
|D| = 1.
either
or
U is collinear
Moreover, these bounds are sharp.
3
In this article we try to generalize this result to the q = ph prime power
case by proving an analogue of Theorem 3 for the case |U| ≤ q. Before
we examine the number of directions determined by less than q affine points
in the plane, we ascend from the plane in the next section and examine the
connection between linear sets and direction sets in arbitrary dimensions.
The further sections will return to the plane.
2
Linear sets as direction sets
The affine space AG(n, q) and its ideal hyperplane Π∞ ∼
= PG(n − 1, q) of
directions together constitute a projective space PG(n, q). We say that the
point P ∈ Π∞ is a direction determined by the affine set U ⊂ AG(n, q) if
there exists at least one line through P that meets U in at least two points.
Definition (Projective linear set). Suppose that GF(s) is a subfield of GF(q).
A projective GF(s)-linear set B of rank d + 1 is a projected image of the
canonical subgeometry PG(d, s) ⊂ PG(d, q) from a center disjoint to this
subgeometry. The projection can yield multiple points.
Proposition 5. Suppose that U is an affine GF(s)-linear set of rank d + 1 in
AG(n, q) such that AG(n, q) is the smallest dimensional affine subspace that
contains U. Let D denote the set of directions determined by U. The set
U ∪ D is a projective GF(s)-linear set of rank d + 1 in PG(n, q) and all the
multiple points are in D.
Proof. Without loss of generality, we can suppose that U contains the origin
and suppose that U is the set of GF(s)-linear combinations of the vectors
a0 , a1 , . . . , ad . We can coordinatize AG(n, q) such that ad−n+1 , . . . , ad is the
standard basis of GF(q)n ∼
= AG(n, q).
Embed GF(q)n ∼
= AG(n, q) into GF(q)d+1 ∼
= AG(d+1, q) such that z0 , z1 , . . . ,
zd−n , ad−n+1 , . . . , ad is the standard basis. Let π denote the projection of
AG(d + 1, q) onto AG(n, q) such that π(zi ) = ai for each i = 0, . . . , d − n
and π(aj ) = aj for each j > d − n. Then U is the image of the canonical
subgeometry AG(d+1, s) by π.
Extend π to the ideal hyperplane. The extended π̄ is a collineation so the
image of a determined direction is a determined direction, and vice versa, let
A and B two arbitrary distinct points in ℓ ∩ U and let P be the direction
determined by π −1 (A) and π −1 (B). Then the direction of ℓ is π̄(P ).
Corollary 6. If D is the set of directions determined by an affine GF(s)linear set, then D is a projective GF(s)-linear set.
4
Remark 7. In [4, Proposition 2.2], Olga Polverino proved that if D is a
projective GF(s)-linear set then |D| ≡ 1 (mod s).
The proposition above says that the set of directions determined by an
affine linear set is a projective linear set. The converse of this proposition is
also true; each projective linear set is a direction set:
Theorem 8. Embed PG(n, q) into PG(n+1, q) as the ideal hyperplane and
let AG(n + 1, q) = PG(n + 1, q) \ PG(n, q) denote the affine part. For each
projective GF(s)-linear set D of rank d + 1 in PG(n, q), there exists an affine
GF(q)-linear set U of rank d + 1 in AG(n+1, q) such that the set of directions
determined by U is D.
Proof. D ⊂ PG(n, q) is the image of the canonical subgeometry PG(d, s) ⊂
PG(d, q) by the projection π : PG(d, q) → PG(n, q) where the center C of
π is disjoint to this subgeometry. Embed PG(d, q) into PG(d + 1, q) as the
ideal hyperplane and extend π to π̄ : PG(d+1, q) → PG(n+1, q) such that its
center remains C. That is, the center is in the ideal hyperplane. Consider
the canonical subgeometry PG(d+1, s) ⊂ PG(d+1, q) and its image by π̄.
π̄
⊂
PG(d+1, s) −−−→ PG(d+1, q) −−−→ PG(n+1, q)






y
y
y
PG(d, s)
⊂
−−−→
PG(d, q)
π
−−−→
PG(n, q)
The ‘ideal part’ of this canonical subgeometry PG(d, s) is the original canonical subgeometry PG(d, s) of PG(d, q) and the projection π̄ project this onto
D. Since the center is totally contained in the ideal hyperplane, π̄ maps the
affine part of the canonical subgeometry PG(d+1, s) one-to-one.
The directions determined by the affine part of PG(d+1, s) are the points of
PG(d, s) in the ideal hyperplane of AG(d+1, q). Since the extended π̄ preserves collinearity, the set of directions determined by the projected image
of the affine part is D.
3
The Rédei polynomial of less than q points
Let U be a set of less than q affine points in AG(2, q) and let D denote the
set of directions determined by U. Let n = |U| and let R(X, Y ) be the
inhomogeneous affine Rédei polynomial of the affine set U, that is,
R(X, Y ) =
Y
(a,b)∈U
n
(X − aY + b) = X +
5
n−1
X
i=0
σn−i (Y )X i
where the abbreviation σk (Y ) means the k-th elementary symmetric polynomial of the set {b − aY | (a, b) ∈ U} of linear polynomials.
Proposition 9. If y ∈ D then R(X, y) ∈ GF(q)[X s(y) ] \ GF(q)[X p·s(y) ].
If y ∈
/ D then R(X, y) | X q − X.
Proof. Assume y ∈ D. Then the equation R(X, y) = 0 has root x with
multiplicity m if there is a line with slope y meeting U in exactly m points.
The value of x determines this line. So each x is either not a root of R(X, y)
or a root with multiplicity a multiple of s(y); and p · s(y) does not have this
property. Since R is totally reducible, it is the product of its root factors.
If y ∈
/ D then a line with direction y cannot meet U in more than one point,
so an x cannot be a multiple root of R(X, y).
Notation. Let F be the polynomial ring GF(q)[Y ] and consider R(X, Y )
as the element of the univariate polynomial ring F[X]. Divide X q − X by
R(X, Y ) as a univariate polynomial over F and let Q denote the quotient
and let H + X be the negative of the remainder.
Q(X, Y ) = (X q − X) div R(X, Y )
−X − H(X, Y ) ≡ (X q − X) mod R(X, Y )
So
q
q
R(X, Y )Q(X, Y ) = X + H(X, Y ) = X +
over F
over F
q−1
X
hq−i (Y )X i
i=0
∗
where degX H < degX R. Let σ denote the coefficients of Q,
q−n−1
Q(X, Y ) = X
q−n
+
X
∗
σq−n−i
(Y )X i
i=0
and so
hj (Y ) =
j
X
∗
σi (Y )σj−i
(Y ).
i=0
We know that deg hi ≤ i, deg σi ≤ i and deg σi∗ ≤ i. Note that the σ ∗ (Y )
polynomials are not necessarily elementary symmetric polynomials of linear
polynomials and if y ∈ D then Q(X, y) is not necessarily totally reducible.
Remark 10. Since degX H < degX R, we have hi = 0 for 1 ≤ i ≤ q − n. By
definition, σ0 = σ0∗ = 1. The equation h1 = 0 implies σ1∗ = −σ1 , this fact and
the equation h2 = 0 implies σ2∗ = −σ2 + σ12 and so on, the q − n equations
hi = 0 uniquely define all the coefficients σi∗ .
6
Proposition 11. If y ∈ D then Q(X, y), H(X, y) ∈ GF(q)[X s(y) ] and if
deg R ≤ deg Q then Q(X, y) ∈ GF(q)[X s(y) ] \ GF(q)[X p·s(y) ].
If y ∈
/ D then R(X, y)Q(X, y) = X q + H(X, y) = X q − X. In this case
Q(X, y) is also a totally reducible polynomial.
Proof. If y ∈ D then
n
R(X, y) = X +
n−1
X
i=0
σn−i (y)X i ∈ GF(q)[X s(y) ] \ GF(q)[X p·s(y) ].
So s(y) | n and σi (y) 6= 0 ⇒ s(y) | n − i ⇒ s(y) | i. The defining equation of
σi∗ contains the sum of products of some σj where the sum of indices (counted
with multiplicities) is i. Since σj (y) 6= 0 only if s(y) | j, also σi∗ (y) 6= 0 only
if s(y) | i.
If deg R ≤ deg Q then we can consider R as (X q − X) div Q and the
reminder is the same H.
Since both R(X, y) and Q(X, y) are in GF(q)[X s(y) ], H(X, y) ∈ GF(q)[X s(y) ].
If y ∈
/ D then R(X, y) | (X q − X) in GF(q)[X] so Q(X, y) is also totally
reducible.
Remark 12. Note that H(X, y) can be an element of GF(q)[X p·s(y) ]. If
H(X, y) ≡ a is a constant polynomial, then R(X, y)Q(X, y) = X q + a =
X q + aq = (X + a)q . This means that R(X, y) = (X + a)n and thus, there
exists exactly one line (corresponding to X = −a) of direction y that contains
U, and so D = {y}.
Definition. If |D| ≥ 2 (i.e. H(X, y) is not a constant polynomial) then for
each y ∈ D, let t(y) denote the maximal power of p such that H(X, y) =
fy (X)t(y) for some fy (X) ∈
/ GF(q)[X p ].
H(X, y) ∈ GF(q)[X t(y) ] \ GF(q)[X t(y)p ].
In this case t(y) < q since t(y) ≤ degX H < q. Let t be the greatest common
divisor of the numbers t(y), that is,
t = gcd t(y) = min t(y).
y∈D
y∈D
If H(X, y) ≡ a (i.e. D = {y}) then we define t = t(y) = q.
Remark 13. If there exists at least one determined direction y ∈ D such
that H(X, y) is not constant then t < q. From Proposition 11 we have
s(y) ≤ t(y) for all y ∈ D, so s ≤ t.
7
Proposition 14. Using the notation above,
R(X, Y )Q(X, Y ) = X q + H(X, Y ) ∈ SpanF h1, X, X t , X 2t , X 3t , . . . , X q i.
Proof. If |D| = {y} then H(X, y) ≡ a and H(X, z) = −X for z 6= y.
Suppose that |D| ≥ 2. If y ∈
/ D then X q + H(X, y) = X q − X and if y ∈ D
then X q + H(X, y) ∈ GF(q)[X t(y) ] \ GF(q)[X t(y)p ].
Thus, in both cases, if i 6= 1 and i ∤ t, then hq−i (Y ) has q roots and its degree
is less than q.
4
Bounds on the number of directions
Although, in the original problem, the vertical direction ∞ was not determined, from now on, without loss of generality we suppose that ∞ is a
determined direction (if not, we apply an affine collineation). We continue
to suppose that there is at least one non-determined direction.
Lemma 15. If ∞ ∈ D $ ℓ∞ then |D| ≥ degX H(X, Y ) + 1.
Proof. If y ∈
/ D then R(X, y) | X q − X, thus H(X, y) = −X and thus
∀i 6= q − 1: hi (y) = 0.
If y ∈ D then R(X, y) ∤ X q − X, hence ∃i 6= q − 1: hi (y) 6= 0 and thus hi 6≡ 0.
Let i be the smallest index such that hi 6≡ 0 and so i = q − degX H. Since
hi 6≡ 0 has at least (q + 1) − |D| = q − (|D| − 1) roots, degY hi ≥ q − |D| + 1.
Now q ≥ deg X q−i hi (Y ) = q − i + degY hi ≥ 2q − |D| + 1 − i.
Hence |D| ≥ q + 1 − i = degX H + 1.
Lemma 16. Let κ(y) denote the number of the roots of X q + H(X, y) in
GF(q), counted with multiplicity. If X q + H(X, y) 6= X q − X and if H(X, y)
is not a constant polynomial, then
κ(y) − 1
κ(y) + t(y)
+1=
≤ t(y) · deg fy (X) = degX H ≤ deg H
t(y) + 1
t(y) + 1
Proof. Fix y ∈ D and utilize that X q + H(X, y) ∈ GF(q)[X t(y) ], thus
t(y)
t(y)
q/t(y)
q
X
+ fy (X)
= X + H(X, y) = a(X) · b(X) · c(X)
where the totally reducible a(X) contains all the roots (in GF(q)) without
multiplicity, the totally reducible b(X) contains the further roots (in GF(q)),
8
and c(X) has no root in GF(q). (Note that t(y) < q so X q/t(y) ∈ GF(q)[X p ].)
)
a(X) | (X q − X)
=⇒ a(X) | fy (X)t(y) + X
q
t(y)
a(X) | X + fy (X)
b(X) | ∂X X q/t(y) + fy (X) = ∂X fy (X)
a(X)b(X) | fy (X)t(y) + X ∂X fy (X)
And so, deg(a(X)b(X)) ≤ t(y) · deg fy + deg fy − 1 = (t(y) + 1) · deg fy − 1,
since ∂X fy (X) 6= 0 and fy (X)t(y) = H(X, y) 6= −X. We get
κ(y) + t(y)
deg(a(X)b(X)) + 1
= t(y)
≤ t(y) · deg fy (X)
t(y) + 1
t(y) + 1
using κ(y) = t(y) · deg(a(X)b(X)).
Using these lemmas above we can prove a theorem similar to Theorem 3
but it is weaker in our case.
Theorem 17. Let U ⊂ AG(2, q) be an arbitrary set of points and let D denote
the directions determined by U. We use the notation s and t defined above
geometrically and algebraically, respectively. Suppose that ∞ ∈ D. One of
the following holds:
either
1=s≤t<q
and
or
1<s≤t<q
and
or
1≤s≤t=q
and
|U| − 1
+ 2 ≤ |D| ≤ q + 1;
t+1
|U| − 1
|U| − 1
+ 2 ≤ |D| ≤
;
t+1
s−1
D = {∞}.
Proof. The third case is trivial (t = q means |D| = 1, by the definition of t).
Let P be a point of U and consider the lines connecting P and the ideal
points of D. Since each such line meets U and has a direction determined by
U, it is incident with U in a multiple of s points. If s > 1 then counting the
points of U on these lines we get the upper bound.
If t < q then we can choose a direction y ∈ D such that the conditions of
Lemma 16 hold. Using Lemma 15 and Lemma 16, we get
|D| ≥ degX H(X, Y ) + 1 ≥
κ(y) − 1
+ 1 + 1.
t(y) + 1
The number of roots of R(X, y)Q(X, y) is at least the number of roots of
R(X, y) which equals to |U|. Thus κ(y) ≥ |U|. And thus
κ(y) − 1
|U| − 1
≥
.
t(y) + 1
t+1
9
An affine collineation converts Szőnyi’s and Blokhuis’ Theorem 4 to the
special case of our Theorem 17, since t is equal to either 1 or p in the case
q = p prime.
In the case q > p, the main problem of Theorem 17 is, that the definition
of t is non-geometrical. Unfortunately, t = s does not hold in general. For
example, let U be a GF(p)-linear set minus one point. In this case s = 1, but
t = p. In the rest of this article, we try to describe this problem.
5
Maximal affine sets
One can easily show that a proper subset of the affine set U can determine the
same directions. (For example, let U be an affine subplane over the subfield
GF(s). Arbitrary s + 1 points of U determine the same directions.)
Definition (Maximal affine set). We say that U ⊆ AG(2, q) is a maximal
affine set that determines the set D ⊆ ℓ∞ ∼
= PG(1, q) of directions if each
affine set that contains U as a proper subset determines more than |D| directions.
Tamás Szőnyi proved a ‘completing theorem’ (stability result) in [6],
which was slightly generalized in [5] as follows.
Theorem 18 (Szőnyi; Sziklai). [5, Theorem 3.1] Let D denote the set of
directions determined by the affine set U ⊂ AG(2, q) containing q − ε points,
√
where ε < α q and |D| < (q + 1)(1 − α), 1/2 < α < 1. Then U can
be extended to a set U ′ with |U ′ | = q such that U ′ determines the same
directions.
Szőnyi’s stability theorem above also stimulates us to restrict ourselves
to examine the maximal affine sets only. (An affine set of q points that does
not determine all directions is automatically maximal.)
If we examine polynomials in one variable instead of Rédei polynomials,
we can get similar ‘stability’ results. Such polynomials occur when we examine R(X, y), Q(X, y) and H(X, y), or R, Q and H over GF(q)(Y ). The second
author conjectured that if ‘almost all’ roots of a polynomial g ∈ GF(q)[X]
have muliplicity a power of p then the quotient X q div g extends g to a
polynomial in GF(q)[X p ]. We can prove more.
Notation. Let p = char F 6= 0 be the characteristic of the arbitrary field F.
Let s = pe and q = ph two arbitrary powers of p such that e ≤ h (i.e. s | q
but q is not necessarily a power of s).
10
Theorem 19. Let g, f ∈ F[X] be polynomials such that g · f ∈ F[X s ]. If
0 ≤ deg f ≤ s − 1 then X q div g extends g to a polynomial in F[X s ].
Proof. We know that deg(gf ) = ks (k ∈ N). Let r = X q mod (f g) denote
the remainder, that is, X q = (gf )h + r where deg r ≤ deg(f g) − 1.
Now we show that h ∈ F[X s ]. Suppose to the contrary that h ∈
/ F[X s ],
s
i.e. the polynomial h has at least one monomial ā ∈
/ F[X ]. Let ā denote
such a monomial of maximal degree. Let b̄ denote the leading term of f g.
Since gf ∈ F[X s ], also b̄ ∈ F[X s ], and since b̄ is the leading term, deg b̄ =
deg(f g). The product āb̄ is a monomial of the polynomial (f g)h, and since
deg b̄ > deg r, then āb̄ is also a monomial of the polynomial (f · g · h + r).
The monomial āb̄ is not in F[X s ], because ā ∈
/ F[X s ] and b̄ ∈ F[X s ]. But
f · g · h + r = X q ∈ F[X s ], which is a contradiction.
Hence r ∈ F[X s ], and so s | deg r, and thus, if the closed interval [deg g,
deg(gf ) − 1] does not contain any integer that is a multiple of s then deg r
is less than deg g.
If we know that deg r < deg g then from the equation X q = (gf )h + r we get
X q = g(f h) + r hence r = X q mod g and X q div g = f · h, where f is a
polynomial such that f g ∈ F[X s ] and also h ∈ F[X s ].
So it is enough to show that the closed interval [deg g, deg(gf ) − 1] does not
contain any integer that is a multiple of s. Using deg(f g) = ks, the closed
interval [deg g, deg(gf ) − 1] = [ks − deg f, ks − 1] and if 0 ≤ deg f ≤ s − 1
then it does not contain any integer which is the multiple of s.
This theorem above suggests that if the product R(X, y)Q(X, y) is an element of GF(q)[Y ][X p·s(y) ] while R(X, y) ∈ GF(q)[Y ][X s(y) ] \ GF(q)[Y ][X p·s(y) ],
then a ‘completing result’ might be in the background. If U is a maximal
affine set then it cannot be completed, so we conjecture the following.
Conjecture 20. If U is a maximal affine set that determines the set D of
directions then t(y) = s(y) for all y ∈ D where t(y) > 2.
Note that there can be maximal affine sets which are not linear.
Example (Non-linear maximal affine set). Let U ⊂ AG(2, q) be a set, |U| =
q+3
q, s = 1, q ≥ |D| ≥
. In this case U cannot be linear because then s
2
would be at least p. But U must be maximal since q + 1 points in AG(2, q)
would determine all directions. Embed AG(2, q) into AG(2, q m ) as a subgeometry. Then U ⊂ AG(2, q m ) is a maximal non-linear affine set of less than
q m points.
But if s > 2, we conjecture that the maximal set is linear.
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Conjecture 21. If U is a maximal affine set that determines the set D of
directions and t = s > 2 then U is an affine GF(s)-linear set.
Although we conjecture that the maximal affine sets with s = t > 2 are
linear sets, the converse is not true.
Example (Non-maximal affine linear set). Let AG(2, si ) be a canonical subgeometry of AG(2, q = si·j ) and let U be an affine GF(s)-linear set in the
subgeometry AG(2, si ) that contains more than si points. Then U determines
the same direction set that is determined by the subgeometry AG(2, si ).
References
[1] S. Ball The number of directions determined by a function over a finite
field. J. Combin. Theory Ser. A 104(2): 341–350, 2003.
[2] A. Blokhuis Personal communication.
[3] A. Blokhuis, S. Ball, A. E. Brouwer, L. Storme and T. Szőnyi
On the number of slopes of the graph of a function defined on a finite
field J. Combin. Theory Ser. A 86(1): 187–196, 1999.
[4] O. Polverino Linear sets in finite projective spaces. Discrete Math.
310(22): 3096–3107, 2010.
[5] P. Sziklai On subsets of GF(q 2 ) with dth power differences. Discrete
Math. 208/209: 547–555, 1999. Combinatorics (Assisi, 1996).
[6] T. Szőnyi On the number of directions determined by a set of points in
an affine Galois plane. J. Combin. Theory Ser. A 74(1): 141–146, 1996.
[7] T. Szőnyi Around Rédei’s theorem Discrete Math. 208/209: 557–575,
1999. Combinatorics (Assisi, 1996).
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